04 fm synthesis

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Copyright 2002-2013 by Roger B. Dannenberg

INTRODUCTION TO COMPUTER MUSIC

FM SYNTHESISRoger B. DannenbergProfessor of Computer Science, Art, and Music

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WEEK 4, DAY 1

Frequency ModulationFM Synthesis

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Frequency ModulationFrequency modulation occurs naturally:

Voice inflection, natural jitter, and vibrato in singing Vibrato in instruments Instrumental effects, e.g. electric guitar Many tones begin low and come up to pitch Loose vibrating strings go sharp as they go louder Slide trombone, Theremin, voice, violin, etc. create

melodies by FM (as opposed to, say, pianos)

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Frequency Modulation with Nyquist fmosc(basic-pitch,fm-control[, table[,phase]])fm-controlis expressed as deviation in Hzhzosc(fm-control)fm-controlis absolute frequency in Hzsnd-compose(f, g) Computes f(g(t)) if g is non-linear, frequency changes

occur

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ExamplesSee Code 4 (code_4.htm)

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Why FM Synthesis?Weve already seen wavetable or table-lookupsynthesis: Very efficient Create any harmonic spectrum Simple frequency and amplitude control

Whats missing? Time-varying control over the spectrum Inharmonic spectra

Various Approaches: Synthesize each sinusoid separately tedious, costly Filter the output of table useful, but only harmonic output FM Synthesis

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FM SynthesisWhen modulation frequency is in the audio range,interesting things happen.

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Carrier Frequency (C)

Modulation Frequency (M)

Number of significant

partials is roughlythe ratio

of modulation amp (freq

dev, D) to modulation freq.

Bandwidth~2(D+M)


Mathematics of FMThe exact amplitudes of the partials generated byFM are described by Bessel functions

These functions are messy, their evolution ismessier, and there is no simple way to invert the

functions

Many lives of FM: 1967-1968 Invented by John Chowning, patented 1975 1983-1986: Yamaha DX7 160,000 sold 1990-1995: IBM PC-compatible Sound Cards 2000s: FM synthesis provides polyphonic ring tones

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FM and HarmonicsGenerated frequencies are:

Where C = Carrier and M = ModulatorSimplest case: C = MGenerated frequencies are:

C+nM gives us C, 2C, 3C, 4C,

What about negative frequencies?

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CnM


FM and Harmonics (2)

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Bandwidth~2(D+M)C

C


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FM and Harmonics (3)

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Bandwidth~2(D+M)C

C


Classic FM brass soundCharacterized by a rise in upper partialsGenerated by increasing depth of modulationUses 1:1 Carrier:Modulation frequency

See example in code_4.htm

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More partials over time


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Odd Harmonics

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Let M = 2CResulting frequencies are C, 3C, 5C, Negative frequencies are -C, -3C, -5C, Try it

CnM


Other Harmonic Schemes

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Let M = i/j x C, for small integers i and jLet F = C/j, then M = iFC = jF, C+M = (i+j)F, C+2M = (2i+j)F, etc.All frequencies are harmonics (integer multiples)of F

Try it

CnM


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Inharmonic Partials

Let M = not i/j x CResulting frequencies are not harmonicsNegative frequencies are not harmonicsTry it

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CnM


FormantsResonances (especially in the vocal tract)emphasize frequencies around the resonant

frequency

We can simulate resonances (and voice) byplacing a carrier near the desired resonant

frequency and modulating it to create nearby

harmonics:

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C

M


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FM Variations SummaryChange C:M ratioPlace carrier at an upper partial to createformant effect

Inharmonic ratios create metalic soundsSee code_4.htm for examples

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Listening FM Examples, Horner, Beauchamp, & Haken FM Examples, track 36, Current Directions, The

importance of periodic and random vibrato in the

perception of vocal tones is demonstrated by adding first

a random and then a periodic vibrato.

FM Examples, track 37, Current Directions, A male vocaltone can be synthesized with a small modification to the

algorithm

Dannenberg, The Side Band from Fanfares forTrumpet and Computer (AM, not FM)

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Project 2 (reminder)

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Use of Samples (Audio Taken From

Other Recordings) Does the sample provide substantial musical material?

Take an existing complete recording and change it with effects bad. Building a drum track from isolated snare samples ok. Processing a song beyond recognition probably ok.

Does your piece offer a commentary on the sample? Irony? Mixing What a Wonderful World with machine gun fire: in principle this would be ok: the song

is obviously a quote and youre making a commentary. But What a Wonderful World was

recorded as a war protest song. Using someone elses irony as your own means the essence of

your work is derivative.

Is the material a quotation or a direct use? If you sample hey baby in the middle of a serene electronic soundscape, thats an obvious

allusion to pop culture, maybe a commentary on commercialism, etc. Obviously a quotation. Its

ok.

If you sample hey baby in the middle of a rap tune, you are leveraging someone elses workand talent to augment your own work and effectively plagiarizing the sample. Not good.

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SummaryFM Synthesis

Time varying spectra Low cost (simplest case is only 2 oscillators) Simple parametric control Musically useful results

FM Control Carrier:Modulator ratio

Harmonic or inharmonic spectra Odd or all harmonics Formants

Depth of modulation Number of partials

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WEEK 4, DAY 2

Behavioral AbstractionFilters

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Temporal Semantics and

Behavioral AbstractionExtensions to ordinary (Lisp, SAL) semantics:

Behaviors Evaluation environment Transformations Temporal combination: SEQ and SIM

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BehaviorsNyquist sound expressions denote a whole classof behaviors

The specific sound computed by the expressiondepends upon the environment

Transformations like STRETCH andTRANSPOSE alter the behavior.

Behaviors vs. linear transformation: when youplay a longer note, you dont simply stretch the

signal! The behavior concept is critical for music.

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Evaluation EnvironmentTo implement behavior concept, allNyquistexpressions evaluate within an environment.

Nyquist environment includes: starting time,stretch factor, transposition, legato factor,

loudness, sample rates, and more.

Environment is hidden and changed oraccessed using special function-like constructs.

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Manipulating the EnvironmentExample:osc(c4) ~ 3

Within STRETCH, all expressions see alteredenvironment and behave accordingly

Scoping is dynamic:function tone() return osc(c4)

play tone() ~ 3


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Manipulating the EnvironmentExample:osc(c4) ~ 3

Within STRETCH, all expressions see alteredenvironment and behave accordingly

Scoping is dynamic:function tone() return osc(c4)

play tone() ~ 3 Transformations can be nested:function tone() return osc(c4) ~ 2

play tone() ~ 3 ICM Week 4 27


Absolute Transformations You can override the inherited environment:function tone2() return osc(c4) ~~ 2

play tone2() ~ 100 Even though TONE2 is called with a stretch factor of 100,

its STRETCH-ABS transformation overrides the

environment and sets it to 2

Once sound is computed by OSC(C4), the soundis immutable, i.e. not subject to transformation!!!!!

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The SOUND Typeosc(c4) ~~ 2 this is an expressionWhen evaluated, osc() uses the environment(especially start time and stretch factor) and

returns a SOUND:

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Start time

Logical stop time

Terminate timeSample Rate


Examplebeginwith x = osc(c4)

play x ~ 3


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Examplebeginwith x = osc(c4)

play x ~ 3end

function x() return osc(c4)play x() ~ 3 !

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Transformations

STRETCH, STRETCH-ABS (~, ~~)AT, AT-ABS (@, @@)LOUD, LOUD-ABSSUSTAIN, SUSTAIN-ABSABS-ENV use default environmentSee manual for others.Maybe well talk about time-varyingtransformations later in semester.

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Practical NotesIn practice, the most critical transformations areAT (@) and STRETCH (~), which control when

sounds are computed and how long they are.

Technically, transformations are not functionsbecause they do not evaluate their arguments in

the normal order: instead, they manipulate the

environment, evaluate the behavior, then restore

the environment.

Implemented as macros in XLISPICM Week 4 33


SEQHow do we make a sequence of sounds:seq(osc(c4), osc(d4))

Semantics: Evaluate osc(c4) at default time (t=0) Resulting sound has logical stop time of 1.0 Evaluate osc(d4) at start time t=1.0 Return the sum of the results

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Counterexample You MUST use seq with behavior expressions, not sound values: set x = osc(c4); compute soundssety = osc(d4);

play seq(x, y) ; WRONG!!

function x() return osc(c4); define

function y() return osc(d4); behaviors

play seq(x(), y()) ; RIGHT!!

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SIMSIMis exactly the same asSUMand +SIMevaluates a list of behaviors and forms theirsum (equivalent to audio mixing)

sim(osc(c4), osc(g4))

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Example Using @play sim(osc(c4),

osc(e4) @ 0.1,

osc(g4) @ 0.2,

osc(b4) @ 0.3),

osc(d5) @ 0.4))

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Overlap With Logical Stop Timesplay seq(set-logical-stop(osc(c4), 0.1),

set-logical-stop(osc(e4), 0.1),

set-logical-stop(osc(g4), 0.1),

set-logical-stop(osc(b4), 0.1),

set-logical-stop(osc(d5), 0.1))

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Logical stop time Physical stop time

Starttim

es


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ScoresWeve seen scores alreadyTo evaluate a score, evaluate each soundexpression with the start time and stretch factor:

{{start dur {instr parameters}} instr(parameters) ~ dur @ start

Note: instr() ~ dur @ start instr() @ (start / dur) ~ dur

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SummarySOUNDS

Start time Logical stop time Physical stop time

Functions evaluated in an environment Dynamically scoped inherited across calls Modified by transformations

Stretch (~) Shift (@)

Results of functions (SOUNDS) are immutableSim and Seq control constructs

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04 fm synthesis

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